Lemma 3 (Hoeffding's inequality [1]}) 
Let \(X_1, \dots , X_n\)  be i.i.d. random variables with bounded intervals: \(a_i\le X_i\le b_i\) , \(i = 1,\dots , n\) . Let \(\bar{X} = \frac{1}{n}\sum _{i=1}^{n}X_i\) , then we have
\(\text{Pr}\left(\left|\bar{X} - \mathrm {E}[\bar{X}]\right| \ge c \right) \le 2\exp \left(-\frac{2n^2c^2}{\sum _{i=1}^{n}(b_i-a_i)^2} \right).\) 

